Multifractal Structure of the Solar Wind

Wiesław M. MacekSpace Research Centre, Polish Academy of Sciences,

Bartycka 18 A, 00-716 Warsaw, Poland;Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszynski University,

Woycickiego 1/3, 01-938 Warsaw, Poland;Dipartimento di Fisica, Universita della Calabria, Rende-Cosenza, Italy.e-mail: macek@cbk.waw.pl, http://www.cbk.waw.pl/∼macek

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Plan of Presentation1. Introduction• Fractals and Multifractals• Solar Wind• Importance of Multifractality

2. Methods of Data Analysis• Magnetic Field Strength Fluctuations• Generalized Scaling Property• Measures and Multifractality• Multifractal Model for Magnetic Turbulence• Degree of Multifractality and Asymmetry

3. Results for Voyager spacecraft• Multifractal Singularity Spectrum• Degree of Multifractality and Asymmetry• Multifractal Turbulence at the Termination Shock

4. Conclusions

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Prologue

A fractal is a rough or fragmentedgeometrical object that can be subdividedin parts, each of which is (at leastapproximately) a reduced-size copy ofthe whole. Fractals are generally self-similar and independent of scale (fractaldimension).

A multifractal is a set of intertwinedfractals. Self-similarity of multifractalsis scale dependent (spectrum ofdimensions). A deviation from astrict self-similarity is also calledintermittency.

Two-scale Cantor set.

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Importance of Multifractality

Starting from seminal works of Kolmogorov (1941) and Kraichnan (1965) many authors haveattempted to recover the observed scaling laws, by using multifractal phenomenological modelsof turbulence describing distribution of the energy flux between cascading eddies at variousscales (Meneveau and Sreenivasan, 1987, Carbone, 1993, Frisch, 1995).

In particular, multifractal scaling of this flux in solar wind turbulence using Helios (plasma)data in the inner heliosphere has been analyzed by March et al. (1996). It is knownthat fluctuations of the solar magnetic fields may also exhibit multifractal scaling laws. Themultifractal spectrum has been investigated using magnetic field data measured in situ byVoyager in the outer heliosphere up to large distances from the Sun (Burlaga, 1991, 1995,2004) and even in the heliosheath (Burlaga and Ness, 2010; Burlaga et al., 2006).

To quantify scaling of solar wind turbulence we have developed a generalized two-scale weighted Cantor set model using the partition technique (Macek 2007; Macek andSzczepaniak, 2008), which leads to complementary information about the multifractal natureof the fluctuations as the rank-ordered multifractal analysis (cf. Lamy et al., 2010).

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We have investigated the spectrum of generalized dimensions and the correspondingmultifractal singularity spectrum depending on one probability measure parameter and tworescaling parameters. In this way we have looked at the inhomogeneous rate of the transferof the energy flux indicating multifractal and intermittent behavior of solar wind turbulence.In particular, we have studied in detail fluctuations of the velocity of the flow of the solarwind, as measured in the inner heliosphere by Helios (Macek and Szczepaniak, 2008),Advanced Composition Explorer (ACE) (Szczepaniak and Macek, 2008), and Voyager in theouter heliosphere (Macek and Wawrzaszek, 2009 ), including Ulysses observations at highheliospheric latitudes (Wawrzaszek and Macek, 2010).

In December 2004 at distances of 94 AU from the Sun Voyager 1 crossed the terminationheliospheric shock separating the Solar System plasma from the surrounding heliosheathand entered the subsonic solar wind, where it encountered quite unusual conditions. It isworth noting that magnetic fields in the heliosheath are normally distributed in contrast to thelognormal distribution in the outer heliosphere (Burlaga et al., 2005). It has also appeared thatthe magnetic field in the near heliosheath, at ∼95 AU, has a multifractal structure (Burlaga etal., 2006).

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The results of the generalized dimensions and the multifractal spectrum obtained using theVoyager 1 data of the magnetic field strength have been discussed at distances of 83.4–85.9AU from the Sun, i.e., before the termination shock crossing (Burlaga, 2004), and in theheliosheath at 94.2–97.2 AU (Burlaga et al., 2006) and 108.5–112.1 AU (Burlaga and Ness,2010), correspondingly. The comparison of the weighted one-scale and two-scale Cantorset models and other formulae fitting the experimental data have been presented in Figures3–6 of the paper by (Macek and Wawrzaszek, 2010), where the dependence on the variousparameters of these models and the resulting degrees of multifractality and asymmetry havealso been thoroughly discussed. In particular, space filling turbulence has been recovered(Burlaga et al., 1993).

The aim of this study is to examine the question of scaling properties of intermittentsolar wind turbulence using our weighted two-scale Cantor set model at a wide range of theheliospheric distances focusing on the solar cycle variations. In particular, we show that thedegree of multifractality modulated by the solar activity is also decreasing with distance: beforeshock crossing is greater than that in the heliosheath. Moreover, we demonstrate that themultifractal spectrum is asymmetric before shock crossing, in contrast to the nearly symmetricspectrum observed in the heliosheath; the solar wind in the outer heliosphere may exhibit strongasymmetric scaling.

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• W. M. Macek, Multifractality and Intermittency in the Solar Wind, Nonlin. Processes Geophys. 14, 695–700(2007).

• W. M. Macek and A. Szczepaniak, Generalized Two-Scale Weighted Cantor Set Model for Solar WindTurbulence, Geophys. Res. Lett., 35, L02108 (2008).

• A. Szczepaniak and W. M. Macek, Asymmetric Multifractal Model for Solar Wind Intermittent Turbulence,Nonlin. Processes Geophys., 15, 615–620 (2008).

• W. M. Macek, Chaos and Multifractals in the Solar System Plasma, in Topics on Chaotic Systems, edited by C.H. Skiadas et al., Selected Papers from Chaos 2008 International Conference, World Scientific, New Jersey,pp. 214–223 (2009).

• W. M. Macek and A. Wawrzaszek, Evolution of Asymmetric Multifractal Scaling of Solar Wind Turbulence inthe Outer Heliosphere, J. Geophys. Res., 114, A03108 (2009).

• W. M. Macek, A. Wawrzaszek, T. Hada, Multiscale Multifractal Intermittent Turbulence in Space Plasmas, J.Plasma Fusion Res. Series, vol. 8, 142-147 (2009).

• W. M. Macek, Chaos and Multifractals in the Solar Wind, Adv. Space Res., 46, 526–531 (2010).• W. M. Macek and A. Wawrzaszek, Multifractal Turbulence at the Termination Shock, Solar Wind 12, American

Institute of Physics, vol. CP1216 (2010), pp. 572–575.• H. Lamy, A. Wawrzaszek, W. M. Macek, T. Chang, New Multifractal Analyses of the Solar Wind Turbulence:

Rank-Ordered Multifractal Analysis and Generalized Two-scale Weighted Cantor Set Model, Solar Wind 12,American Institute of Physics, vol. CP1216 (2010), pp. 124–127.

• A. Wawrzaszek and W. M. Macek, Observation of Multifractal Spectrum in Solar Wind Turbulence by Ulyssesat High Latitudes, J. Geophys. Res., 115, A07104 (2010).

• W. M. Macek and A. Wawrzaszek, Multifractal Structure of Small and Large Scales Fluctuations ofInterplanetary Magnetic Fields, Planet. Space Sci., 59, 569–574 (2011a).

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• W. M. Macek and A. Wawrzaszek, Multifractal Two-Scale Cantor Set Model for Slow Solar Wind Turbulence inthe Outer Heliosphere During Solar Maximum, Nonlin. Processes Geophys., 18, 287-294 (2011b).

• W. M. Macek, A. Wawrzaszek, V. Carbone, Observation of the Multifractal Spectrum at the Termination Shockby Voyager 1, Geophys. Res. Lett., 38, L19103 (2011).

• W. M. Macek, Multifractal Turbulence in the Heliosphere, in Exploring the Solar Wind, edited by M. Lazar,Intech, ISBN 978-953-51-0399-4 (2012).

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Fractal

A measure (volume) V of a set as afunction of size l

V (l)∼ lDF

The number of elements of size l neededto cover the set

N(l)∼ l−DF

The fractal dimension

DF = liml→0

lnN(l)ln1/l

Multifractal

A (probability) measure versus singularitystrength, α

pi(l) ∝ lαi

The number of elements in a small rangefrom α to α + dα

Nl(α)∼ l− f (α)

The multifractal singularity spectrum

f (α) = limε→0

liml→0

ln[Nl(α+ ε)−Nl(α− ε)]

ln1/l

The generalized dimension

Dq =1

q−1liml→0

ln∑Nk=1(pk)

q

ln l

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Multifractal Characteristics

Fig. 1. (a) The generalized dimensions Dq as a function of any real q, −∞ < q < ∞,and (b) the singularity multifractal spectrum f (α) versus the singularity strength α withsome general properties: (1) the maximum value of f (α) is D0; (2) f (D1) = D1; and (3)the line joining the origin to the point on the f (α) curve where α = D1 is tangent to thecurve (Ott et al., 1994).

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Generalized Scaling Property

The generalized dimensions are important characteristics of complex dynamical systems; theyquantify multifractality of a given system (Ott, 1993). In the case of turbulence cascadethese generalized measures are related to inhomogeneity with which the energy is distributedbetween different eddies (Meneveau and Sreenivasan, 1991). In this way they provideinformation about dynamics of multiplicative process of cascading eddies. Here high positivevalues of q> 1 emphasize regions of intense fluctuations larger than the average, while negativevalues of q accentuate fluctuations lower than the average (cf. Burlaga 1995).

Using (∑ piq ≡ 〈pi

q−1〉av) a generalized average probability measure

µ(q, l)≡ q−1√〈(pi)q−1〉av (1)

we can identify Dq as scaling of the measure with size l

µ(q, l) ∝ lDq (2)

Hence, the slopes of the logarithm of µ(q, l) of Eq. (2) versus log l (normalized) provides

Dq = liml→0

log µ(q, l)log l

(3)

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Measures and Multifractality

Similarly, we define a one-parameter q family of (normalized) generalized pseudoprobabilitymeasures (Chhabra and Jensen, 1989; Chhabra et al., 1989)

µi(q, l)≡pq

i (l)

∑Ni=1 pq

i (l)(4)

Now, with an associated fractal dimension index fi(q, l) ≡ logµi(q, l)/ log l for a given q themultifractal singularity spectrum of dimensions is defined directly as the average taken withrespect to the measure µ(q, l) in Eq. (4) denoted by 〈. . .〉

f (q) ≡ liml→0

N

∑i=1

µi(q, l) fi(q, l) = liml→0

〈logµi(q, l)〉log(l)

(5)

and the corresponding average value of the singularity strength is given by(Chhabra and Jensen, 1987)

α(q) ≡ liml→0

N

∑i=1

µi(q, l) αi(l) = liml→0

〈log pi(l)〉log(l)

. (6)

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Methods of Data Analysis

Magnetic Field Strength Fluctuations and Generalized Measures

Given the normalized time series B(ti), where i = 1, . . . ,N = 2n (we take n = 8), to each intervalof temporal scale ∆t (using ∆t = 2k, with k = 0,1, . . . ,n) we associate some probability measure

p(x j, l)≡1N

j∆t

∑i=1+( j−1)∆t

B(ti) = p j(l), (7)

where j = 2n−k, i.e., calculated by using the successive (daily) average values 〈B(ti,∆t)〉 of B(ti)between ti and ti +∆t. At a position x = vswt, at time t, where vsw is the average solar windspeed, this quantity can be interpreted as a probability that the magnetic flux is transferred to asegment of a spatial scale l = vsw∆t (Taylor’s hypothesis).

The average value of the qth moment of the magnetic field strength B should scale as

〈Bq(l)〉 ∼ lγ(q), (8)

with the exponent γ(q) = (q−1)(Dq−1) as shown by Burlaga et al. (1995).

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Mutifractal Models for Turbulence

Fig. 1. Generalized two-scale Cantor set model for turbulence (Macek, 2007).

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Solutions

Transcendental equation (for n→ ∞)

pq

lτ(q)1

+(1− p)q

lτ(q)2

= 1 (9)

For l1 = l2 = λ and any q in Eq. (9) one has for the generalized dimensions

τ(q)≡ (q−1)Dq =ln[pq+(1− p)q]

ln λ. (10)

Space filling turbulence (λ = 1/2):the multifractal cascade p−model for fully developed turbulence,the generalized weighted Cantor set (Meneveau and Sreenivasan, 1987).The usual middle one-third Cantor set (without any multifractality):p = 1/2 and λ = 1/3.

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Degree of Multifractality and Asymmetry

The difference of the maximum and minimum dimension (the least dense and most densepoints in the phase space) is given, e.g., by Macek (2006, 2007)

∆≡ αmax−αmin = D−∞−D∞ =

∣∣∣∣log(1− p)log l2

− log(p)log l1

∣∣∣∣. (11)

In the limit p→ 0 this difference rises to infinity (degree of multifractality).

The degree of multifractality ∆ is simply related to the deviation from a simple self-similarity.That is why ∆ is also a measure of intermittency, which is in contrast to self-similarity (Frisch,1995, chapter 8).

Using the value of the strength of singularity α0 at which the singularity spectrum has itsmaximum f (α0) = 1 we define a measure of asymmetry by

A≡ α0−αmin

αmax−α0. (12)

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Schematic of the Heliospheric Boundaries.

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Fig. 2. The heliospheric distances from the Sun and the heliographic latitudes during eachyear of the Voyager mission. Voyager 1 and 2 spacecraft are located above and below the solarequatorial plane, respectively.

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Voyager Spacecraft

7 – 60 AU (1980 – 1995)70 – 90 AU (1999 – 2003)95 – 107 AU (2005 – 2008)

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Fig. 1. The multifractal singularity spectrum of the magnetic fields observed by Voyager 1 (a) in the solar windnear 50 and 90 AU (1992, diamonds, and 2003, triangles) and (b) in the heliosheath near 95 and 105 AU (2005,diamonds, and 2008, triangles) together with a fit to the two-scale model (solid curve), suggesting change of thesymmetry of the spectrum at the termination shock (Macek et al. 2011).

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Fig. 2. The degree of multifractality ∆ in the heliosphere versus the heliospheric distances compared to a periodically decreasing

function (dotted) during solar minimum (MIN) and solar maximum (MAX), declining (DEC) and rising (RIS) phases of solar cycles, with

the corresponding averages shown by continuous lines. The crossing of the termination shock (TS) by Voyager 1 is marked by a vertical

dashed line. Below is shown the Sunspot Number (SSN) during years 1980–2008 (Macek et al. 2011).

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Fig. 3. The degree of asymmetry A of the multifractal spectrum in the heliosphere as a function of the heliosphericdistance during solar minimum (MIN) and solar maximum (MAX), declining (DEC) and rising (RIS) phases of solarcycles with the corresponding averages denoted by continuous lines; the value A = 1 (dotted) corresponds to theone-scale symmetric model. The crossing of the termination shock (TS) by Voyager 1 is marked by a verticaldashed line (Macek et al. 2011).

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Figure 1: Degree of multifractality ∆ and asymmetry A for the magnetic fieldstrengths in the outer heliosphere and beyond the termination shock (Macek etal. 2011).

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Conclusions

• Using our weighted two-scale Cantor set model, which is a convenient tool to investigate theasymmetry of the multifractal spectrum, we confirm the characteristic shape of the universalmultifractal singularity spectrum. f (α) is a downward concave function of scaling indices α.

• For the first time we show that the degree of multifractality for magnetic field fluctuations ofthe solar wind falls steadily with the distance from the Sun and seems to be modulated bythe solar activity.

• Moreover, we have considered the multifractal spectra of fluctuations of the interplanetarymagnetic field strength before and after shock crossing by Voyager 1. In contrast to theright-skewed asymmetric spectrum with singularity strength α > 1 inside the heliosphere,the spectrum becomes more left-skewed, α < 1, or approximately symmetric after the shockcrossing in the heliosheath, where the plasma is expected to be roughly in equilibrium in thetransition to the interstellar medium.

• We also confirm the results obtained by Burlaga et al. (2006) that before the shockcrossing, especially during solar maximum, turbulence is more multifractal than that in theheliosheath.

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• In addition, outside the heliosphere during solar minimum the spectrum seems to bedominated by values of α < 1. This is very interesting because it represents a first directin situ information of interest in the astrophysical context beyond the heliosphere. In fact,a density of measure dominated by α < 1 implies that the magnetic field in the very localinterstellar medium is roughly confined in thin filaments of high magnetic density.

• That is the heliosheath is possibly more dominated by ’voids’ of magnetic fields, thusimplying that magnetic turbulence tends to be more ’passive’ (cf. Frisch, 1995) in the verylocal interstellar medium.

• These informations, obtained in situ rather than through scintillations observations, arerelevant in the context of interstellar turbulence confirming stellar formation modeling (e.g.,Spangler et al. 2009), related to the presence of very localized intense magnetic fieldstructures.

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Epilogue

Within the complex dynamics of the solarwind’s and interstellar medium fluctuatingintermittent plasma parameters, there isa detectable, hidden order described bya Cantor set that exhibits a multifractalstructure.

This means that the observedintermittent behavior of magneticfluctuations in the heliosphere and thevery local interstellar medium may resultfrom intrinsic nonlinear dynamics ratherthan from random external forces.

Thank you!

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This work was supported by the European Community’s Seventh FrameworkProgramme ([FP7/2007–2013]) under Grant agreement No. 313038/STORM andcosponsored by the Ministry of Science and Higher Education (2013–2015), Poland(MNiSW).

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